Reciprocity Theorem

The reciprocity theorem is a fundamental property of linear, bilateral networks that describes a symmetric relationship between excitation and response. It finds direct application in antenna theory, two-port network analysis, and GATE problems involving source-response interchange.

Core Concept Explanation

The reciprocity theorem states that in a linear, bilateral network, if a voltage source Vs applied at one port produces a current I at another port, then the same voltage source Vs when moved to the second port will produce the same current I at the first port. The ratio of response to excitation remains unchanged upon interchange.

The theorem applies only to networks containing linear elements (resistors, linear inductors, linear capacitors) and bilateral elements (elements that behave identically for current in either direction). Networks with dependent sources generally do not satisfy reciprocity unless specifically constructed to do so.

Physically, this symmetry arises because bilateral elements have symmetric impedance matrices. For a two-port network described by its Z-parameter matrix, reciprocity is equivalent to saying Z12 equals Z21. This is a direct consequence of the symmetry in Maxwell's equations for passive media.

Mathematical Expression

Consider a two-port network. Let Vs be a voltage source applied at port 1 and I2 be the resulting short-circuit current at port 2. Then let Vs be applied at port 2 and I1 be the short-circuit current at port 1. Reciprocity states:

V1 / I2 = V2 / I1, which simplifies to Z12 = Z21 in the Z-parameter representation. Equivalently, in Y-parameters, Y12 = Y21. The transfer impedance from port 1 to port 2 equals the transfer impedance from port 2 to port 1.

For a general network with N nodes, if a source Vs is placed between nodes A and B and the current through a branch between nodes C and D is measured, then placing Vs between C and D and measuring current between A and B gives the same result. This generalizes the theorem beyond the simple two-port view.

Practical Understanding

In antenna theory, reciprocity means the radiation pattern of a transmitting antenna equals its receiving pattern. This is extremely powerful because calculating one pattern is sufficient. In filter design, reciprocity guarantees that a passive LC filter has the same insertion loss whether the signal travels forward or backward through it.

For GATE problems, reciprocity is most commonly tested using simple resistive networks. Students are given a source-response pair and asked to find the current after the source is moved to the response location. The answer is always the same current, and the trap is forgetting that conditions of the original circuit (like the short circuit at the response port) must be maintained.

Solved Numerical Example

A voltage source of 10V is connected between nodes A and B in a resistive network. A short-circuit current of 2A is measured between nodes C and D. The same 10V source is now connected between C and D while A-B is short-circuited. Find the current through the A-B branch.

Given:
Vs = 10V (source voltage)
I_CD = 2A (short-circuit current at C-D when source is at A-B)

Why this formula applies:
The network is linear and bilateral (purely resistive), so the reciprocity theorem holds.
The transfer impedance Z_transfer = Vs / I_response is the same in both directions.

Formula:
Z12 = Z21  =>  Vs / I_CD = Vs / I_AB

Substitution:
10 / 2 = 10 / I_AB

Calculation:
I_AB = 10 / (10/2) = 10 / 5 = 2A

Final Answer: I_AB = 2A
The current at A-B equals the original 2A, confirming reciprocity.

Exam Tip: Reciprocity applies only to linear bilateral networks with independent sources. Networks with dependent sources or active elements do NOT satisfy reciprocity. Always check this condition before applying the theorem in GATE problems.

Mechanism in Summary

• Reciprocity holds when the network is linear and all elements are bilateral (no diodes, transistors, or dependent sources).
• The excitation-response ratio, called transfer impedance, remains constant regardless of which port the source is placed at.
• In Z-parameter form, reciprocity is expressed as Z12 = Z21, making the Z matrix symmetric.
• When the source is moved from port 1 to port 2, the original response location must be short-circuited to maintain consistency of measurement.
• The theorem is used in antenna systems, filter analysis, and simplifying complex resistive network problems in GATE.

Quick Revision

• Reciprocity theorem: In a linear bilateral network, swapping source and response positions gives the same response magnitude.
• Applicable to: Resistors, linear L and C, independent sources only. NOT applicable to networks with dependent sources.
• Z-parameter condition: Z12 = Z21 (symmetric Z matrix). Y-parameter condition: Y12 = Y21.
• Transfer impedance: Zt = Vs / I_response is the same in both directions.
• Exam trap: The short-circuit condition at the response port must be maintained when source is moved. Forgetting this changes the problem entirely.
• Physical basis: Symmetry in Maxwell's equations for passive, isotropic, linear media.
• Application: Antennas (transmit = receive pattern), passive LC filters (bidirectional insertion loss equality).